Reference Energies for Non-Relativistic Core Ionization Potentials

Abstract: Deep-lying core electrons carry highly localized, site-specific information that forms the basis of X-ray photoelectron spectroscopy. Accurately predicting their associated core ionization potentials (IPs) is a demanding theoretical task, requiring a balanced treatment of strong orbital relaxation, electron correlation, and relativistic effects. Over the years, a variety of approaches have been developed, ranging from state-specific wave function methods to linear-response formalisms and Green's function techniques. However, their assessment has often relied on comparisons with experiment, where multiple sources of error (basis set incompleteness, relativistic corrections, and vibrational effects) are entangled, making it difficult to isolate the performance of correlation treatments. In the present work, we establish a consistent, theory-based benchmark for core IPs by computing 84 non-relativistic values (73 second-row and 11 third-row IPs) at the full configuration interaction level within the core-valence separation approximation, using large correlation-consistent basis sets augmented with tight-core and diffuse functions (aug-cc-pCVXZ). These results define theoretical best estimates within a fixed finite basis set, providing a chemically accurate reference for method development and validation. Importantly, our dataset allows for systematic, theory-versus-theory comparisons that disentangle correlation and relaxation effects from other physical contributions. On this basis, we assess the performance of widely used approximate methods, including equation-of-motion coupled-cluster approaches up to the inclusion of quadruple excitations, the one-shot $G_0W_0$ scheme, as well as state-specific methods.

• The paper establishes a benchmark of 84 core ionization potentials using CVS-FCI to isolate errors in correlation and relaxation.
• It systematically evaluates methods including CC hierarchies, state-specific Δ approaches, and G0W0 schemes to determine accuracy in core IP predictions.
• The results offer a critical reference for computational spectroscopy, guiding both theoretical advancements and basis set optimizations for XPS interpretation.

Reference Energies for Non-Relativistic Core Ionization Potentials: A Technical Perspective

Introduction and Motivation

The prediction of molecular core ionization potentials (IPs)—energies associated with removing core electrons—rests at the core of computational spectroscopy, supporting interpretability of X-ray photoelectron spectroscopy (XPS) measurements in chemistry and materials science. Accurate theoretical descriptions of core-ionized states demand treating strong orbital relaxation, dynamic correlation, and relativistic effects with balanced, high-level quantum chemical approaches. However, benchmarking the accuracy of approximate methods is confounded by systematic experimental uncertainties, which include basis set incompleteness, omitted relativistic/vibrational corrections, and core-hole meta-stability/continuum couplings.

This paper establishes a computational benchmark of 84 non-relativistic core IPs (73 second-row and 11 third-row IPs) computed at the CVS full configuration interaction (CVS-FCI) level in large correlation-consistent basis sets (aug-cc-pCVXZ, $X=$ D, T, Q) with explicit tight-core and diffuse augmentation. The primary aim is to isolate the intrinsic accuracy of correlation and relaxation treatments by providing chemically accurate, theory-only reference values for method development and theory-to-theory comparisons.

Methodology

Reference Calculations

Core IPs are calculated using the core-valence separation full configuration interaction (CVS-FCI) approach:

• Reference State Optimization: The core-ionized determinants use restricted open-shell HF (ROHF) orbitals optimized using the maximum overlap method (MOM), ensuring proper core-hole localization.
• Selection of Determinants: The selected CI (CIPSI) method is employed, with determinant spaces carefully truncated and subsequently expanded using natural orbitals for improved convergence.
• CVS Approximation: The determinant space is restricted to configurations with at least one hole in the targeted core orbital (CVS), an approach rooted in the well-established decoupling of bound-core and continuum-ionized states.
• Extrapolation: Energies are statistically extrapolated to the FCI limit using weighted regression on the second-order perturbative correction.

Calculations utilize the aug-cc-pCVXZ basis, with results reported primarily at the triple-zeta (ACVTZ) and, where feasible, quadruple-zeta levels.

Approximate Methods

Systematic benchmarking is performed for several classes of methods:

• Equation-of-Motion Coupled-Cluster (EOM-CC) Series: CC2, CCSD, CC3, CCSDT, CC4, CCSDTQ, all with CVS where applicable.
• State-Specific $\Delta$ Methods: $\Delta$SCF, $\Delta$MP2, and $\Delta$CCSD, with both ROHF and UHF reference determinants.
• Green's Function Approach (G0W0): One-shot G0W0 based on hybrid PBE functionals with variable exact exchange (45% for second-row, 70% for third-row elements), following recommendations for improved core IPs.

Composite basis set schemes are employed for high-cost CC4/CCSDTQ calculations, leveraging the additivity of excitation energy increments across basis sets and excitation ranks.

Numerical Benchmarks and Key Results

• Invariance and Dependence on Orbitals: The CVS-FCI energies show weak dependence on the choice of underlying orbitals (optimized vs. neutral HF), with deviations below 0.02 eV, confirming the robustness of state-specific orbital optimization in the CVS context.
• Systematic Method Evaluation:
  • CC Hierarchy: MAEs decrease monotonically from 2.05 eV (CCSD) to 0.05 eV (CCSDTQ), demonstrating systematic improvability. At the CCSDTQ level, residual errors occasionally reach 0.1 eV, underscoring the persistent challenge of capturing core-hole relaxation within linear-response frameworks.
  • State-Specific Methods: $\Delta$ROHF achieves errors comparable to CC3 (MAE of 0.57 eV), with UHF-based methods displaying systematic underbinding (MSE $\sim -0.5$ eV). $\Delta$UMP2 and $\Delta$UCCSD do not surpass linear-response counterparts due to limitations in relaxation treatment at the MP2 level.
  • G0W0 Schemes: For second-row elements, G0W0@PBEh(45) achieves MAEs of 0.48 eV, but for third-row atoms, only G0W0@PBEh(70) yields well-behaved solutions (MAE 0.51 eV), indicating strong dependency of G0W0 on the starting point for core states.
• Atom-Specific Trends:
  • Errors in approximations grow with atomic number: fluorine core IPs systematically yield the largest MAEs for all methods (barring the highest-accuracy methods or those with compensating basis set effects), correlating with increased relativistic contributions at higher $Z$.
• Experimental Contextualization:
  • Theoretical CVS-FCI results in ACVTZ agree closely with experiment for second-row elements due to fortuitous error compensation (typical deviations $\Delta$0 0.1 eV), but substantial discrepancies are present for third-row atoms (deviations $\Delta$1 6 eV), dominated by missing scalar and spin-orbit relativistic corrections.

Practical and Theoretical Implications

This dataset provides—essentially for the first time—a systematic theoretical reference for core IPs, controlling for all but correlation and relaxation errors within a fixed finite basis. This enables:

• Method Evaluation and Development: Automated, high-throughput benchmarking of emerging quantum chemical methods (e.g., coupled-cluster with perturbative quadruples, multi-reference Green's function approaches) can now use these reference IPs to dissect and optimize correlation vs. relaxation approximations without contamination by experimental noise or extraneous physical effects.
• Basis Set Optimization: The data supply a controlled foundation for developing and testing compact or hybrid basis sets specifically tailored for core-ionized states and XPS simulations.
• Guidance for XPS Interpretation: Although direct experiment-theory comparisons require further relativistic/vibrational corrections, the present reference energies serve as a critical anchor point for indirect benchmarking, calibration, and error attribution.

Future directions include calculation of transition intensities for direct comparison with experimental spectra (necessitating additional methodological development within the state-specific SCI context), as well as extension to satellite states that encode correlation-induced shake-up effects.

Conclusion

By delivering internally consistent, high-accuracy reference values for 84 non-relativistic core IPs across the second and third-row elements, this work provides a critical benchmark for both theoretical spectroscopy and quantum chemistry method development. The systematic assessment of EOM-CC, state-specific, and Green's function methods reveals their respective strengths and limitations in capturing core-level spectroscopy. The database, as an extension of the quest dataset, anchors future evaluations of both canonical and emerging electronic structure methods and underpins future methodological development for high-accuracy XPS simulation and interpretation.